Explain and apply empirical Bayesian estimation in the nonparametric and semiparametric cases.

Empirical Bayes is what you do when the Bühlmann formulas demand a structural mean and variance you do not actually know. You estimate them from the loss history itself, then plug into $Z=n/(n+k)$ like nothing happened.

Why empirical Bayes exists. Pure Bühlmann credibility needs three structural quantities: the collective mean $\mu =E[\mu (\Theta )]$, the expected process variance $v=E[\text{Var}(X|\Theta )]$, and the variance of hypothetical means $a=\text{Var}(\mu (\Theta ))$. In practice you observe only loss data across $r$ policyholders. Empirical Bayes replaces $\mu$, $v$, and $a$ with sample-based estimators $\hat{\mu }$, $\hat{v}$, $\hat{a}$, then forms $\hat{k}=\hat{v}/\hat{a}$ and the credibility factor.

KEY: Empirical Bayes does NOT estimate the prior distribution of $\Theta$. It estimates only the two moments $v$ and $a$ that the credibility premium needs.

• Nonparametric: no assumption about the conditional distribution $f(x|\theta )$. You estimate $v$ directly from within-policyholder variation.
• Semiparametric: you assume a specific conditional family (typically Poisson for claim counts), which links $v$ to $\mu$ and saves a degree of estimation effort.

Common mistakes

• Dividing the pooled within-sum-of-squares by $rn$ instead of $r(n-1)$. The correct denominator counts degrees of freedom inside each group.
• Forgetting to subtract $\hat{v}/n$ from the between-variance when computing $\hat{a}$. The raw between-variance has expectation $a+v/n$, not $a$.
• Reporting a negative $\hat{a}$ as the answer. Floor it at 0 and set ${\hat{Z}}_i=0$.

Bottom line

• Nonparametric EB estimates EPV $v$ and VHM $a$ directly from data with no assumption on the conditional distribution.
• Equal sample sizes $n$: $\hat{v}$ is the pooled within-group variance with denominator $r(n-1)$; $\hat{a}$ is the between-group variance minus $\hat{v}/n$.
• Bühlmann-Straub (unequal exposures $m_{ij}$) replaces simple means with exposure-weighted means and gives each policyholder ${\hat{Z}}_i=m_i/(m_i+\hat{k})$.
• Bühlmann-Straub $\hat{a}$ uses divisor $m-\sum m_i^2/m$, not $r-1$.

Exam shortcut

Equal exposures and equal $n$: grand mean is the simple average of class means. Skip the weighted arithmetic. Bühlmann-Straud denominator shortcut: $m-\sum m_i^2/m=\sum m_i(1-m_i/m)$. Useful when there are only 2 or 3 groups. For Poisson semiparametric, write $\hat{v}=\bar{X}$ FIRST, then plug into the $\hat{a}$ formula. Do not compute a pooled within-variance you do not need.

The full lesson (about 2,151 words, 14 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 5c

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